Question

In how many maximum different ways can 3 identical balls be placed in the 12 squares (each ball to be placed in the exact centre of the squares and only one ball is to be placed in one square) shown in the figure given below such that they do not lie along the same straight line?

(a) 144 (b) 200 (c) 204 (d) 216

Answer 1

(b) 3 balls can be placed be in any of the 12 squares in ¹²C₃ ways. 

Total number of arrangements = ¹²C₃ = 220 Now, assume that balls lie along the some line. There can be 3 cases : 

Case I: When balls lie along the straight horizontal line. 

3 balls can be put in any of the 4 boxes along the horizontal row in ⁴C₃ ways. 

Now, since there are 3 rows, so number of ways for case I = ⁴C₃ × 3 = 12 

Case II : When balls lie along the vertical straight line 3 balls can be put in any of the 3 boxes along the vertical row in ³C₃ ways.

 Now, as there are 4 vertical rows, so number of ways for Case II = ³C₃ × 4 = 4 

Case III : Balls lie along the 2 diagonal lines towards the left and 2 diagonal lines towards the right. Number of ways = 2 + 2 = 4 

Number of ways, when balls lie along the line = 12 + 4 + 4 = 20 

Number of ways when balls don't lie along the line = Total number of ways – number of ways when balls lie along the line. = 220 – 20 = 200.